The study of shock waves is an important issue in the theory of both inviscid and viscous conservation laws, such as the compressible Euler equations and N-S equations in gas dynamics.
We first overview several remarkable works in one dimension.
The shock stability for hyperbolic conservation laws has been done by many articles such as [1]}, [2]}.
In the viscous case, based on a comparison principle, the study was initiated by [1]} for the scalar conservation laws. Then [4]}, [5]} extended the stability results of viscous shocks to the systems by using basic energy methods, provided that the initial perturbations satisfy a zero-mass condition.
Without this condition, the problems with generic perturbations were solved by introducing additional diffusion waves and coupled diffusion waves propagating along the transverse characteristics; see [6]}, [7]} for the uniformly parabolic systems.
And [8]}, [9]} made more subtle analysis to study some physical models including the compressible N-S equations. We also refer to [10]}, [11]} and the references therein for the spectral stability of viscous shocks, where the shock-strength is allowed to be large, and to [12]}, [13]}, [14]} for the nonlinear shock stability, which can be derived from the spectral stability in some appropriate senses.
